Lecture 2 Hypothesis Test Dr. Hoda Ragab Rezk

Methods of Evaluating Tests (1) Powerfulness (2) Unbiasedness and Invariancy (3) Local Powerfulness In order to examine some of these criteria, some terminologies such as error probabilities, power functions, type I error, and type II error are needed .

Definition Let 𝐻 0 : 𝜃∈ Ω 0 and 𝐻 𝑎 : 𝜃∉ Ω 0 be the null and alternative hypothesis to be tested based on a random sample X1,X2, ...,Xn from a population X with density f(x; 𝜃), where 𝜃 is a parameter. The significance level of the hypothesis test 𝐻 0 : 𝜃∈ Ω 0 and 𝐻 𝑎 : 𝜃∉ Ω 0 denotes by 𝛼, is defined as 𝛼=P (Type I Error)

This is also equivalent to The significance level of a hypothesis test we mean the probability of rejecting a true null hypothesis, that is 𝛼=P (Reject H 0 | H 0 is True) This is also equivalent to 𝛼=P (Accept H a | H 0 is True)

β=P (Accept H 0 | H 0 is flase) Definition Let 𝐻 0 : 𝜃∈ Ω 0 and 𝐻 𝑎 : 𝜃∉ Ω 0 be the null and alternative hypothesis to be tested based on a random sample X1,X2, ...,Xn from a population X with density f(x; 𝜃), where 𝜃 is a parameter. The probability of type II error of the hypothesis test 𝐻 0 : 𝜃∈ Ω 0 and 𝐻 𝑎 : 𝜃∉ Ω 0 denotes by β, is defined as β=P (Accept H 0 | H 0 is flase)

β=P (Accept H 0 / H a is false) This is also equivalent to β=P (Accept H 0 / H a is false)

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